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Incommensurate atomic and magnetic modulations in the spin-frustrated beta-NaMnO2 triangular lattice

Citation for published version:Orlandi, F, Aza, E, Bakaimi, I, Kiefer, K, Klemke, B, Zorko, A, Arcon, D, Stock, C, Tsibidis, GD, Green, MA,Manuel, P & Lappas, A 2018, 'Incommensurate atomic and magnetic modulations in the spin-frustratedbeta-NaMnO2 triangular lattice', Physical Review Materials, vol. 2, no. 7, 074407.https://doi.org/10.1103/PhysRevMaterials.2.074407

Digital Object Identifier (DOI):10.1103/PhysRevMaterials.2.074407

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1

Incommensurate atomic and magnetic modulations in the spin-frustrated

-NaMnO2 triangular lattice.

Fabio Orlandi,1 Eleni Aza,2,3 Ioanna Bakaimi,2,† Klaus Kiefer,4 Bastian Klemke,4 Andrej Zorko,5

Denis Arčon,5,6 Christopher Stock,7 George D. Tsibidis,2 Mark A. Green,8 Pascal Manuel,1 and

Alexandros Lappas2,*

1 ISIS Facility, Rutherford Appleton Laboratory, Harwell Oxford, Didcot OX11 0QX, United Kingdom

2 Institute of Electronic Structure and Laser, Foundation for Research and Technology–Hellas, Vassilika

Vouton, 71110 Heraklion, Greece 3 Department of Materials Science and Engineering, University of Ioannina, 451 10 Ioannina, Greece

4 Department Sample Environment and CoreLab Quantum Materials, Helmholtz-Zentrum Berlin für

Materialien und Energie GmbH, D-14109 Berlin, Germany 5 Jozef Stefan Institute, Jamova c. 39, 1000 Ljubljana, Slovenia

6 Faculty of Mathematics and Physics, University of Ljubljana, Jadranska c. 19, 1000 Ljubljana, Slove-

nia 7 School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom

8 School of Physical Sciences, University of Kent Canterbury, Kent CT2 7NH, United Kingdom

ABSTRACT. The layered β-NaMnO2, a promising Na-ion energy-storage material has been investigat-

ed for its triangular lattice capability to promote complex magnetic configurations that may release

symmetry restrictions for the coexistence of ferroelectric and magnetic orders. The complexity of the

neutron powder diffraction patterns underlines that the routinely adopted commensurate structural mod-

els are inadequate. Instead, a single-phase superspace symmetry description is necessary, demonstrating

that the material crystallizes in a compositionally modulated q= (0.077(1), 0, 0) structure. Here, Mn3+

Jahn-Teller distorted MnO6 octahedra form corrugated layer stacking sequences of the β-NaMnO2 type,

which are interrupted by flat sheets of the α-like oxygen topology. Spontaneous long-range collinear

antiferromagnetic order, defined by the propagation vector k= (½, ½, ½), appears below TN1= 200 K.

Moreover, a second transition into a spatially modulated proper-screw magnetic state (kq) is estab-

lished at TN2= 95 K, with an antiferromagnetic order parameter resembling that of a two-dimensional

(2D) system. The evolution of 23

Na NMR spin-lattice relaxation identifies a magnetically inhomogene-

ous state in the intermediate T-region (TN2 <T< TN1), while its strong suppression below TN2 indicates

that a spin-gap opens in the excitation spectrum. High-resolution neutron inelastic scattering confirms

that the magnetic dynamics are indeed gapped (Δ~5 meV) in the low-temperature magnetic phase,

while simulations on the basis of the single-mode approximation suggest that Mn-spins residing on ad-

jacent antiferromagnetic chains, establish sizable 2D correlations. Our analysis points that novel struc-

tural degrees of freedom promote, cooperative magnetism and emerging dielectric properties in this

non-perovskite-type of manganite.

2

I. INTRODUCTION.

Devising cost-efficient chemical routes for multiferroic magnetoelectric compounds that foster cou-

pling between spins and other electron degrees of freedom is a fascinating problem of both fundamental

and technological interest.1 Engineering the materials’ structure to accommodate unusual coordinations

of interacting neighbours offers one such viable, but challenging avenue. The perturbation of exchange

interactions that emerge from competition due to magnetic frustration, 2,

3,

4 can select complex spin

arrangements that release symmetry restrictions and realize the long-wanted coupling of otherwise mu-

tually exclusive ferroelectric and magnetic orders. In this context, the non-perovskite, two-dimensional

(2D) Na-Mn-O oxides are investigated as a testing ground for such a kind of magnetoelectricity. These

are rock-salt derivatives of the family A+Me

3+O2 (A= alkali metal, Me= 3d transition metal) delafossites

5, 6 that have attracted considerable interest due to their physical and chemical properties. They include,

transparent conducting oxides, such as the CuAlO2 7, superconductors, like the hydrated variant

Na0.3CoO2 ·1.3H2O 8of the P2-NayCoO2 bronzes

9, multiferroics as AFeO2 (A= Na, Ag)

10, 11, and ca-

thodic materials for high-capacity Na-ion re-chargeable batteries, like P2-NayMn1-xMxO2 (x, y ≤ 1, M =

Ni, Mg, Li) 12

. Such intercalation materials show high structural flexibility upon alkali metal insertion

or extraction and give rise to a rich phase diagram. 13

The crystal chemistry of AMeO2 allows for poly-

morphism due to oxygen-layer gliding processes.14

Consequently, their performance is mediated by

phase transitions between nearly degenerate structural types (e.g. designated, as O3- (3R; R-3m) and

P2- (P63/mmc)), 12,15

while extended defects (e.g. stacking faults) formed between various crystal do-

mains, render the apparently simple AxMeO2 bronzes metastable. Therefore, new insights on the impact

of their inherent compositional variation are sought in order to explain their complicated sequences of

electronic and structural processes.

Core concepts of materials science point out that when near-degenerate energy states are involved,

compositional modulation16

often emerges as a naturally evolving process that relieves frustration by

satisfying the cation-anion chemical requirements, as for example in ferroelectrics, 17

and shape

memory alloys. 18, 19

Then, alternatives to traditional crystallographic approaches are necessary in order

to understand how subtle structural modulations in correlated transition metal oxides (e.g. cation order

and tilting of metal-oxygen coordination polyhedral etc.) entangle their electron degrees of freedom and

lead to novel behaviour, extending from heterogeneous catalysis and spin-induced ferroelectricity to

high-temperature superconductivity. The ability to control such functional properties, often emerging in

the framework of broken symmetries (as in TbMnO3 and Ni3V2O8 magnetoelectric materials), 20

relies

in understanding the role of residual disorder governing the modulation of atomic positions and magnet-

ic moments. The superspace formalism, previously implemented for the description of modulated

chemical crystal structures,21

has grown as a powerful method especially when nuclear and magnetic

modulations intertwine in the same phase. 22

Diverse structural types, ranging from perovskites

(CaMn7O12 23

, Pb2MnWO6 24

) to wolframite-type (MnWO4 25

) modulated structures, which all display

symmetry-allowed coupling of electric polarization and magnetization, are illustrative examples of the

importance of a robust and efficient treatment of the symmetry of nuclear and magnetic modulations.

The focus here is on two particular polymorphs in the Na-Mn-O system which crystallizes in distorted

variants of the O3-NaFeO2 structure (3R polytype, R-3m). 26

In these layered compounds the spontane-

ous deformation of the MnO6 octahedra is caused by the Jahn-Teller effect, inherent to the high-spin

Mn3+

cations (

; S= 2; μeff 4.9 μΒ). Because of this distortion, α-NaMnO2 becomes monoclinic

(C2/m), with flat 27

MnO6 sheets (Figure S1a) 28, whilst β-NaMnO2 appears to adopt an orthorhombic

cell (Pmmn), entailing zig-zag 29

MnO6 sheets (Figure S1b) 28

. The latter polytype is similar to the

thermodynamically stable lithiated analogue β-LiMnO2, 30

an important precursor phase for cathode

materials in solid-state Li-ion batteries. 31

Moreover, specific challenges facing the Mn-containing sys-

tems are governed, (a) by the very similar free-energies of the α- and β- NaMnO2 polymorphs, 32

which

3

suggest that intermediate phases with compositional modulations could be formed at a very low energy

cost, and (b) by the Mn topology (see Figure S1, Supplemental Material) 28

that maps out a triangular

lattice 33

, inferring some degree of spin-frustration that renders these polymorphs sensitive to small per-

turbations.

In view of the former characteristic, transmission electron microscopy and synchrotron X-ray powder

diffraction have shown that on the basis of superspace formalism, planar defects could act as a struc-

ture-directing mechanism in the cation-ordered rock-salt-type AMeO2 structures, and in particular, the

α- and β- phases of NaMnO2 can be gradually transformed into each other by changing the density of

the involved twin planes. 33

Interestingly, the presence of local intergrowths of β-polymorph and stack-

ing faults within the lattice of the parent α-NaMnO2 phase is shown to be controlled in single-crystals

grown under optimal conditions.34

This apparent energy degeneracy between α- and β- type oxygen co-

ordinations seems to play an important role in determining the particularly high charge capacity (ca.

190 mA h g−1

) of polycrystalline β-NaMnO2 as an earth-abundant Na-ion cathode. 35

As of the second

inherent feature, neutron powder diffraction has shown that despite the considerable spin-frustration in

α-NaMnO2, Néel order sets-in at 45 K. 36

With this concomitant symmetry breaking, a spin-gap due to

leading quasi-one dimensional interactions (with a predominant nearest-neighbor exchange interaction

of J1~ 72 K37

and frustrated J2 0.44 J1 38

; Figure S1a 28

) describes the low-energy magnetic dynamics,

while a peculiar magnetostructural inhomogeneity emerges as a consequence of the system’s tendency

to remove magnetic degeneracy due to spin frustration. 39,

40

On the other hand, the magnetic ground

state of β-NaMnO2 is less well understood from the experimental point of view. Theoretical calculations

though, predict that a spin-model with two-dimensional couplings (J1~ 70 K nearest neighbor and J3~

57 K next nearest neighbor; Figure S1b) 28

and a weaker frustrated interaction (J2~ 13 K) are likely to

describe the experimental magnetic susceptibility. This material also manifests an abundant quasi-

periodic arrangement of defects. 33

Moreover, room-temperature 23

Na solid-state nuclear magnetic reso-

nance (NMR) spectra supported by first-principles DFT computations identified a wealth of local struc-

tural rearrangements, entailing a trade-off between the majority β-type nanodomains and those of the α-

like phase upon electrochemical cycling of sodium.41

The present contribution provides a new powerful neutron powder diffraction insight on β-NaMnO2,

highlighting that this challenging material is stabilized by near-equivalent in energy lattice confor-

mations. The strength of superspace formalism has been utilized to describe the structure on the basis of

a single-phase model, entailing an incommensurate compositional modulation. The latter is depicted as

a coherent intergrowth of two types of NaMnO2 layers, reflecting the α- and β- type oxygen

coordinations, and is shown to determine the material’s physical properties. We illustrate the implica-

tions of the modified lattice topology, with its intrinsic extended defects, on the successive magnetic

phase transitions. Furthermore, temperature-dependent 23

Na NMR and inelastic neutron scattering ex-

periments point that the magnetic dynamics are gapped, while the influence of the magnetic order on

the electric dipole order is also reflected in the temperature- and field- dependent magnetocapacitance

studies.

II. EXPERIMENTAL METHODS.

Polycrystalline β-NaMnO2 samples were synthesized by a high-temperature solid-state chemistry pro-

tocol reported before, 33

while phase identification was undertaken by x-Ray powder diffraction

(XRPD) experiments carried out on a Rigaku D/MAX-2000H rotating Cu anode diffractometer. β-

NaMnO2 specimens were air-sensitive and all post-synthesis handling was carried with the aid of an Ar-

circulating MBRAUN anaerobic glove box.

4

Dc magnetic susceptibility as a function of temperature (5 ≤ T ≤ 300 K) was measured on 20 mg

batches of powder samples with a Superconducting Quantum Interference Device (SQUID) magnetom-

eter (Quantum Design, MPMS-XL7) under a moderate magnetic field (H = 20 mT). Heat capacity (C)

was measured at zero-field on a cold-pressed pelletized powder sample by means of the relaxation

technique, utilizing a physical property measurement system (Quantum Design, PPMS).

NMR measurements on the 23

Na nucleus (nuclear spin I= 3/2) were performed on a powder sample

sealed in a pyrex sample holder. 23

Na NMR spectra and spin-lattice relaxation rate 1/T1 were recorded

between 50 and 300 K in a magnetic field of 8.9 T using a solid-echo and inversion recovery pulse se-

quences, respectively. Wide-line 23

Na NMR powder data were obtained as sums of individual spectra

acquired by changing the measurement frequency in 50 kHz steps over ±3 MHz around the 23

Na refer-

ence frequency, ν0= 100.5234 MHz, which was determined from a 0.1 M NaCl solution. The spin-

lattice relaxation rate measurements were performed at the position of the central line.

Neutron powder diffraction data were collected on the WISH diffractometer, 42

operating at the second

target station (TS2) at the ISIS pulsed neutron source in the UK. WISH, with its high-brilliance, is par-

ticularly optimized for providing high resolution at long d-spacing required for magnetic studies. For

this purpose, a 2.7 g polycrystalline sample was loaded in a 8 mm V-can, which was then sealed with

indium wire inside a high-quality, He-circulating anaerobic glove box. An Oxford Instrument liquid he-

lium cryostat was used for the temperature dependent diffraction experiments. Data analysis was per-

formed by using the Jana2006 software 43

for the Rietveld refinements, whereas the group theory analy-

sis was performed with the help of the ISODISTORT software. 44

Inelastic neutron scattering work was performed on the MARI direct geometry chopper spectrometer

(ISIS, UK) and also on the DCS spectrometer (NIST, USA). Experiments on MARI used incident ener-

gies Ei=85 and 150 meV, with a Gd Fermi chopper spun at 300 and 450 Hz, respectively. Measurements

on DCS were done with an incident energy of Ei=14.2 meV. A 7.3 g of a powder sample was loaded in

an annular aluminum sachet that was placed inside a cylindrical Al-can for the ISIS experiment, while a

5 g sample was loaded in V-can for the NIST experiment. In either case the cans were sealed with indi-

um wire and they were cooled at low temperatures with a top-loaded closed-cycle refrigerator. All data

has been corrected for background and also phonons from the structural lattice. For the MARI data, the

background plus phonon contribution to the scattering at each energy transfer was estimated from the

high angle detector banks where magnetic scattering is suppressed owing to the Mn3+

form factor. We

have fit the high angle and high momentum detectors at a fixed energy transfer to the form L(Q)= L0 +

L1 × Q2 , with L0 capturing the background and L1 providing an estimate of the phonon scattering. L(Q)

was then used to estimate the background and phonon scattering at low momentum transfers and then it

was subtracted. For data taken on DCS, the background was estimated by using the requirement for de-

tailed balance as discussed previously.45

The dielectric permittivity of ~3 mm pellets of pressed polycrystalline samples, without electrodes at-

tached on the two flat surfaces, was studied at the CoreLab for Quantum Materials in the Helmholtz-

Zentrum, Berlin, with a 14 T PPMS system. The home-made setup is tailored for dielectric constant

measurements in a capacitor-like arrangement. It gives the possibility to select between an AH 2700A

Ultra-precision Capacitance Bridge, for relatively low-frequencies (50 Hz - 20 kHz) or a Solatron 1260

Impedance/Gain Phase Analyser, for the high-frequency region up to 32 MHz; the latter is being used

together with a 1296A Dielectric Interface System in order to cope with ultra-low capacitance levels. A

Lakeshore 370 temperature controller was utilized to cover a broad temperature range (5 ≤T≤ 180 K).

5

III. RESULTS AND DISCUSSION

A. Macroscopic Properties.

The temperature dependent magnetic susceptibility, χ(T), of the different NaMnO2 polymorphs, quali-

tatively appears similar, with exception of the presence of a broad maximum (200 K for α–polytype),36

which apparently shifts to higher temperature in the β–phase (Figure 1a). Such a broad feature is a gen-

eral characteristic of low-dimensional antiferromagnetic systems. However, from χ(T) data alone no

evidence for a transition to a long-range ordered state is observed.

On the other hand, the heat capacity, C(T), measured in zero magnetic field displays several very weak

anomalies (Fig. 1), possibly of magnetic origin. In order to highlight these features, we first estimated

the phonon contribution to the specific heat, Cph(T), and subtracted it from the experimentally measured

heat capacity. Here, Cph(T) assumes a sum of Debye contributions (2< T< 280 K), following the proce-

dure used before for other low-dimensional spin systems: 46,47

(1)

with R (8.314 J mol−1

K−1

) the gas constant,

the Debye temperature and

, while fit-

ting was based on an optimization approach using the minimum number of free parameters. In our case,

the Cph(T) was approximated by two Debye functions, addressing the relatively different atomic masses

of the constituent element-coupled vibrations (cf. Na-O and Mn-O) in the β-NaMnO2. This yielded the

fitting parameters, C1= 0.55(2), C2= 2.0(2) and

,

(Figure 1a). The

vanishingly small magnitude of C(T) at very low temperatures, in accord with the β-phase insulating

nature, agrees well with the T3 term that corresponds to phonons (eq. (1)).

The outcome of the subtraction of Cph(T) from the total heat capacity is shown in Figure 1b. As the

corresponding anomalies in the differential C(T) are very small, pointing to some sensitivity to the de-

fects in the lattice structure (vide-infra), and the estimated phonon part uncertainties are high, they ren-

der further analysis to assess the differential C(T) as a likely magnetic contribution, Δ

, unfavorable at this stage. The identification, though, of the two fairly broad humps cen-

tered at 95 K (TN2) and 200 K (TN1), would suggest that β-NaMnO2 undergoes two transitions. These

qualitative C(T) characteristics therefore require further study to inquire the role of magnetic interac-

tions in such phase changes.

B. 23

Na NMR Dynamics near the Transitions.

A critical aspect of many macroscopic thermodynamic properties is the role of the material’s micro-

scopic dynamical response. Techniques capable of detecting spin dynamics on a local scale, such as sol-

id-state 23

Na NMR, can therefore be helpful to understand the complex behavior of β-NaMnO2. The 23Na NMR powder spectra of β-NaMnO2 were measured between room temperature and 50 K, where

they become very broad and, consequently, the signal becomes very weak and difficult to measure (Fig-

ure 2a). At 300 K, the spectrum has a characteristic powder line shape for a quadrupole I= 3/2 nuclei

with the quadrupole asymmetry parameter of η 0. A closer inspection of the satellite (3/2 1/2)

transitions of the 300 K spectrum terminated around Q = 1.28 MHz from the narrow central transi-

tion (1/2 1/2) line (upper inset to Figure 2a) shows that the expected singularity is rounded, which

is consistent with a high-degree of Na local site disorder. Here Q is the 23

Na quadrupole frequency. On

6

cooling below TN1, there is almost no change of the central transition line. However, a close inspection

of the 23

Na NMR satellite line reveals that a shoulder starts to gradually broaden well beyond Q. This

is clearly seen as a growth of the NMR signal intensity on both sides of the satellite shoulder (lower in-

set of Figure 2a). As the positions of the satellite shoulder remain nearly at the same frequency, the

quadrupole frequency must also remain the same through the transition at TN1. This suggests that no

structural deformation takes place in the vicinity of the Na site, corroborating that the high temperature

transition (TN1) is magnetic in origin. Moreover, below TN1 the intensity of the sharp central peak multi-

plied by temperature (to counterbalance the changing Boltzmann population), starts to progressively

decrease with decreasing temperature below TN1 (Figure 2b). The broadening of the NMR line beyond

the satellites can thus be attributed to growing internal magnetic fields at certain Mn ion sites, while the

gradual wipeout of the central line below TN1 (Figure 2b) reveals that the high-temperature paramagnet-

ic-like signal vanishes only gradually, as it remains present at all temperatures below TN1. This leads us

to the important conclusion that the magnetic state below TN1 is inhomogeneous. On further cooling be-

low TN2, the 23

Na NMR line shape broadening becomes really pronounced as the spectrum becomes

completely dominated by the broad distribution of internal (hyperfine) magnetic fields and the sharp

central peak almost disappears. These line shape changes verify that β-NaMnO2 indeed undergoes two

successive transitions to magnetically ordered states, at 200 K and 95 K, in agreement with the as-

signment of subtle peaks in the differential C(T) as magnetic transitions (Fig. 1b).

Additional information about the two magnetic transitions is deduced from the 23

Na spin-lattice relax-

ation rate 1/T1, which was determined from fitting of 23

Na magnetization recovery curves (Figure 3a) to

the magnetic-relaxation model for I=3/2, 48

. (2)

Here, s < 1 accounts for imperfect inversion of 23

Na nuclear magnetization after the initial pulse,

while stands for a stretching exponent. In the high-temperature paramagnetic (PM) regime, 1/T1 is

nearly temperature independent, 1/T1 = 35(1) s-1

(Figure 3b). Such temperature independence is in fact

anticipated for an exchange-coupled antiferromagnetic (AFM) insulator in the paramagnetic phase. The

stretching exponent is = 0.88 (Figure 3a); a value slightly below 1 implying a small distribution of

relaxation rates expected in experiments on powder samples. The transition to the magnetic state at TN1

is accompanied by a sizeable step-like increase in the 1/T1 value to 1/T1 = 66(5) s-1

and a gradual reduc-

tion of the stretching exponent (Figure 3a). The latter indicates that the distribution of the spin-lattice

relaxation times suddenly starts increasing below TN1 thus indicating growing magnetic inhomogeneity

between TN1 and TN2 which is in accord with the line shape changes (Figure 2a). In fact, as two-step

magnetization-recovery curves are clearly observed below TN1 (e.g., measurement taken at 100 K

shown in Figure 3a), the fit of the magnetization recovery curves in the T1 experiment is significantly

improved if two relaxation components are included. Here, the relative intensity of one of the compo-

nents (AFM1) increases at the expense of the second PM component, the latter in close analogy to the

wipeout effect of the narrow central line (Figure 2b).

We stress that no obvious critical fluctuations leading to diverging 1/T1 could be detected at TN1. The

likely reason is the nature of magnetic fluctuations, which according to the expression,

(where Aq denotes the hyperfine coupling of the 23

Na nuclei with the elec-

tronic magnetic moments, is the imaginary part of the dynamical susceptibility and is the Larmor

frequency), could be filtered out in the 1/T1 measurements for highly symmetric Na (octahedral) sites.

On the other hand, on approaching the lower transition temperature at TN2, the 1/T1 of the paramagnetic

PM component is rapidly enhanced, suggesting the onset of critical fluctuations. A phenomenological

fit of the critical model 1/T1=A+B(T-TN2)-p

to the PM data in the temperature range between TN2 and

7

110 K, yields the critical exponent p = 0.45(10) for A = 66(5) s-1

and TN2 = 95.0(5) K (Figure 3b). Such

critical enhancement demonstrates that the magnetic fluctuations that govern the transition at TN2 can-

not be filtered out anymore at the Na site. This is also consistent with the observed dramatic 23

Na NMR

line shape changes (Figure 2a). The temperature dependence of the other component (AFM1), which

we attribute to the already magnetically ordered regions in the sample, is much more subtle (Figure 3b).

Finally, at T < TN2, the two components in the magnetization recovery curves are not obvious any more

(Figure 3a), so we resort back to a single-exponential fit (eq. 2). However, a very low stretching expo-

nent of 0.34 has to be employed. Such a strikingly low value of indicates an extremely broad distribu-

tion of relaxation times, hence a broad distribution of local magnetic environments below TN2. At the

same time 1/T1 is strongly suppressed below TN2 and exhibits an activated type of dependence ( , Figure 3b), indicating the opening of an excitation gap, , in the low-temperature

phase.

C. Crystallographic structure

Critical to understanding such transformations is the way magnetic ions are arranged in the underlying

lattice structure that establishes nearest-neighbor exchange terms and stabilizes non-degenerate ground

states. High quality data collected on the WISH diffractometer enables the analysis of the crystallo-

graphic structure of β-NaMnO2. The main reflections of the neutron powder diffraction (NPD) pattern

are consistent with the Pmmn space group, with cell parameters ao= 4.7851(2) Å, bo= 2.8570(8) Å, co=

6.3287(4) Å, at 300 K. The Rietveld refinement of the main nuclear reflections (300 K), with the Pmmn

model 29

(Mn1 in 2b position z= 0.617(5), Na1 in 2b z= 0.125(4), O1 in 2a z= 0.365(6) and O2 in 2a z=

0.872(6)), suggests a significant degree of “anti-site” defects between the Mn and Na sites that leads to

an average occupation of 80:20 (see Figure S2, Supplemental Material) 28

. Moreover, the refinement

points to an unexpectedly large-value for the oxygen thermal parameter (Uiso~0.038(2) Å2). The use of

anisotropic temperature factors in the refinement results in a clear elongation of the thermal ellipsoids

along the c-direction (see Figure S2) 27

indicating strong positional disorder. Following this suggestion

we spilt the two oxygen positions along the c-axis and the refinement converged to a splitting of 0.5 Å

and 70:30 occupancy of the resultant sites, with normal isotropic thermal parameters (Uiso~0.015(2)

Å2). It is worth stressing that the split and especially the occupancy of O1 and O2 resemble the anti-site

occupancy of the Mn and Na atoms; in particular, as shown in Figure S2 this distortion is needed to sat-

isfy the coordination requirements of the Na and Mn cations.

A crucial feature of the 300 K NPD pattern, in association with the above analysis, is the presence of

additional reflections that could be ascribed to a nuclear modulation (Figure 4). In support to this comes

earlier transmission electron microscopy (TEM) work on β-NaMnO2 33

, where it was pointed out that

formation of planar defects establishes short-ranged ordered regions that locally (i.e. on the length scale

of a few unit cells) follow the stacking sequence of NaMnO2 layers characteristic of either the α- or the

β- phases. Importantly, long-period stacking sequences, with a modulation vector q=(00) 0.1 (con-

sistent with the cell choice reported in the present work), were also required for indexing the additional

satellite peaks observed in both electron and synchrotron X-ray diffraction data. From a LeBail fit of

the WISH data we obtained an optimal modulation vector q = (0.077(1),0,0), accounting for satellites

up to the second order in the NPD pattern. Some small satellite reflections, however, are sliding off the

calculated position (Fig. 4), suggesting that the other two components of the modulation vector may be

slightly different from zero. Refinements where the other two components of q were allowed to vary

proved unstable and did not lead to reasonable results. The obtained value of q is near the commensu-

rate 1/13 position, which explains why the 1/6 value used before in the synchrotron X-ray diffraction

patterns indexed well a large number of satellite peaks.

8

The observation of the satellite reflections in both NPD and TEM measurements and the refinement of

the average nuclear structure, indicate the possibility of a compositional modulation in the structure that

can be modelled through the superspace formalism. 49, 50

The theory of (3 + D) superspace groups, in-

troduced by de Wolff (1974, 1977), 51, 52

is widely used to describe the symmetry of commensurate and

incommensurate modulated structures. In order to understand the NPD pattern of WISH we therefore

used a (3+1)-dimensional superspace approach considering an occupational modulation for all the sites

in the average nuclear structure. In order to derive the possible superspace groups we performed group

symmetry analysis with the help of the ISODISTORT Software Suite 44

starting from the refined aver-

age structure and the propagation vector q=(00). Having taken into account the observed reflection

conditions and the symmetry properties of the modulation vector, the symmetry analysis led to the

Pmmn(α00)000 superspace group as the best solution, corresponding to the 1 irreducible representa-

tions (IRs), with order parameter direction (OPD) P(,0). 53

To account for the compositional modulation a step-like (Crenel) function is introduced for every site

in the structure. The Crenel function is defined as 54

(3)

where x4 is the internal (fourth) coordinate in the (3+1)D approach and Δ is the width of the occupa-

tional domain centered at x04 (Δ corresponds also to the average fractional occupancy of the site). The

modulation functions on the same cation site are constrained to be complementary, meaning that in eve-

ry point of the crystal the site is occupied (this results in the equations Δ[Mni]+Δ[Nai]=1 and

x4[Mni]=1-x4[Nai] for each cation site). For the split oxygen positions we introduce a similar constraint,

imposing that in any position in the crystal we have the superposition of the two split sites. Regarding

the origin along the fourth axis, the superspace group constrains this value to two equivalent values: 0

and 0.5, thus making the choice trivial. Moreover, an additional constraint is introduced regarding the

two Mn/Na sites. The electron diffraction measurements, reported by Abakumov et al. 33

suggest that

the quasi-periodic stacking sequences of the NaMnO2 layers entail coherent stacking faults, a feature

which points that their modeling can be reduced to the alternation sequence of the Na and Mn cations.

We followed a similar approach for the modeling of the NPD pattern assuming that the step-like func-

tions were constrained to have in every NaMnO2 plane the right Mn/Na ordering, that is to say, when

one site switches from Mn to Na the other changes from Na to Mn. The crystallographic model built in

this way was employed for qualitative Rietveld refinements. Broad, asymmetric reflections throughout

the NPD pattern, mainly due to defects (e.g. stacking faults) and strain make such analysis hard to op-

timize, raising the agreement factors and making a quantitative refinement difficult. The Rietveld plot,

over a wide d-spacing range, is shown in Figure 4 and the associated reliability factors are, Rp= 8.81%,

Rwp= 12.73%, Rmain= 9.96%, Rsat±1= 15.41%, Rsat±2= 14.79%. Despite the apparent reflection broaden-

ing, our model shows good agreement for the modulated parts of the profile, especially obvious in the

relatively short d-spacing region of the pattern (inset in Figure 4). The crystallographic parameters of

the compositionally modulated β-NaMnO2 at 300 K, on the basis of a (3+1)D Rietveld analysis with the

Pmmn(00)000 superspace group (a= 4.7852(4) Å, b= 2.85701(8) Å, c= 6.3288(4) Å, =0.077 (1)) are

compiled in Table S1 28

.

This single-phase structural model, despite the presence of low intensity reflections ascribable to a

small amount of the α-phase and MnO (Fig. 4), takes into account almost all the satellites present in the

NPD pattern of the β-phase, as compared to the two-phase description on the basis of the B2/m(αβ0)00

superspace group derived before from the analysis of the synchrotron X-ray powder diffraction data 33

.

The nuclear structure model obtained here is shown in Figure 5. This is consistent with the one pro-

posed by Abakumov et al.,33

entailing coherent intergrowth of stacking sequences of NaMnO2 layers

along the a0-axis, characteristic of the α- and β- polytypes. It may be considered as good approximation

to the real chemical phase, as planar defects, seen by electron microscopy, could violate the idealized

9

Crenel-type function used in the present analysis of the NPD data. In this model, the MnO6 octahedra

throughout the structure display strong Jahn-Teller distortion (see Figure S3 28

, for oxygen-cation dis-

tances in the (3+1)D approach), with four short bonds below 2 Å and two long ones around 2.4 Å, in a

fashion analogous to the α-NaMnO2 36

. On the other hand, while Na is also octahedrally coordinated to

oxygen, the distances involved are longer due to its larger ionic radius. Moreover, in an effort to visual-

ize the degree of compositional modulation in the β-NaMnO2 structure, Fourier maps of the observed

structure factor (Figure 6) involving the atomic sites in the zx4-plane were computed on the basis of the

observed NPD intensities and the calculated phases. Figure 6a shows the complementary occupation of

the cation sites without any particular modulation of the z coordinate. On the contrary, from the Fourier

maps centered at the oxygen positions (Figure 6b) it is inferred that the site-splitting observed in the

average structure is needed in order to satisfy the coordination requirement of the Mn3+

Jahn-Teller ac-

tive cation. In fact, it is noted that when the Na and Mn swap sites (cf. compositional modulation), the

same happens in the oxygen split positions so that the bonding requirements are restored as depicted in

Figure S3 28

. Our approach demonstrates that having taken advantage of the superspace formalism to

describe the compositional modulation of the Mn and Na sites in a single-phase atomic configuration,

the incommensurate β-NaMnO2 structure can be depicted as a coherent intergrowth of two types of

NaMnO2 layers, reflecting the α- and β- polytype oxygen coordinations (Fig. 5).

D. Magnetic Structure Evolution.

In view of the complex nuclear modulated structure observed in the NPD profiles of β-NaMnO2 it is

challenging to evaluate the correlation between the crystal and magnetic structures as the sample tem-

perature is lowered. The temperature evolution of the diffraction pattern demonstrates the presence of

two magnetic transitions (Figure 7).

First, below TN1 200 K there is an intensity increase at magnetic Bragg peak positions corresponding

to a propagation vector k=(½ ½ ½) with respect to the Pmmn orthorhombic average structure. These

reflections grow quickly below the magnetic transition temperature and their broad Lorentzian-like pro-

file is an indication that the magnetic domain is sensitive to the strain and defects present in the nuclear

structure (refer to Figure 6), complying with the broadening of 23

Na NMR spectra (inset, Figure 2a).

Moreover below about 100 K the diffraction patterns show the development of additional reflections

(Figure 7a). This new set of peaks can be indexed assuming the combination of the magnetic propaga-

tion vector k and the nuclear one q, giving magnetic intensity at the positions hkl[kq]. It is worth

noting that the temperature dependence of the integrated intensity (Figure 7b) of these two sets of re-

flections possesses different critical behavior, thus suggesting that the two magnetic orders likely fall

into different universality classes. In particular the fit of the ½½½ reflection with power law I= I0 [1-

(Τ/ΤΝ)]2β gives a critical exponent of β= 0.33(4), indicating interactions of a 3D nature, instead, the kq

satellites possess an exponent of β= 0.15(8), which is more consistent with 2D interactions (Figure 7b).

Careful analysis of the diffraction pattern reveals the presence of some additional low intensity reflec-

tions that are not indexed with the previous propagation vectors. These extra reflections are ascribed to

a small-content of MnO impurity and the α-polymorph.

Let us first discuss the important changes in the NPD pattern that were observed below 200 K. In or-

der to establish the possible magnetic space group we performed magnetic symmetry analysis with the

help of the ISODISTORT software. 44

The NPD patterns show that no clear magnetic intensity is ob-

served on the nuclear satellite reflections, therefore pointing that the magnetic structure is not strongly

related to the nuclear modulation at least in the 100 < T< 200 K temperature range. For this reason,

magnetic symmetry analysis was initiated on the basis of parent average Pmmn nuclear structure (Fig-

ure S1, Table S1) 28

and the propagation vector k= (½ ½ ½). The results of the symmetry analysis are

10

reported in Table S2 28

. The best agreement between observed and calculated patterns was obtained for

the mR1 representation, with order parameter direction (OPD) P1(a,0), corresponding to the magnetic

space group Ca2/c, with a change in the unit cell with respect to the parent structure described by the

transformation matrix {(0,-2,0),(0,0,2),(-1,1,0)}. It is worth underlining that the space group Ca2 also

gives a reasonably good result (Table S2), but with an increased number of refinable variables, thus

suggesting the higher symmetry option Ca2/c as the best solution. Combining the mR1 P1(a,0) IRs with

the compositional modulated structure, the Ca2'/c'(a0γ)00 magnetic superspace group is obtained. With

the latter we then carried out Rietveld refinements, with the representative 100 K profile. The Rietveld

plot is shown in Figure 8, and the refined parameters are compiled in Table S3 28

. The associated relia-

bility factors are, RFobs= 8.46% for the nuclear reflections and RFmag= 12.50% for the magnetic ones,

while the RP= 13.88%. Their values are rather on the high side, due to pronounced hkl-dependent

broadening, likely arising from the presence of planar defects. The magnetic structure is drawn in Fig-

ure 9, projected in the same plane as the nuclear one (Figure 5, top panel). It entails

antiferromagnetically coupled Mn-chains running down the bo-axis (ao , bo and co setting is with respect

to the orthorhombic Pmmn unit cell), stacked in a zig-zag fashion when viewed in an a0c0-plane projec-

tion (Figure 9a) that gives rise to antiferromagnetically coupled, corrugated MnO2 layers (Figure 9b). A

similar collinear spin-model has been utilized before for the description of the magnetic state in the

isomorphous β-LiMnO2, where three-dimensional long-range order is established at TN 260 K.55

The derived spin-configuration for β-NaMnO2, though, indicates a commensurate ordering only for the

Mn2-site, as a similar ordering on the Mn1 site would have generated strong magnetic intensity at the

nuclear satellite reflections, a case that is not supported by the NPD data. In this compositionally modu-

lated nuclear structure, between 100 < T< 200 K only the NaMnO2 layer stacking sequences character-

istic of the β- polytype carry a net magnetic moment. Such a magnetically inhomogeneous state is con-

sistent with the wipeout of the central 23

Na-NMR line (Fig. 2b) and the two-component nuclear spin-

lattice relaxation in the same temperature range. The magnetic moment of Mn2 sites has been computed

as μ 2.38(10) μB at 100 K, but as the observed NPD profile shows fairly broad magnetic peaks, the

attained staggered moment may be an underestimate (cf. the full moment for spin-2 Mn3+

is expected

to be 4 μB).

When temperature is lowered below TN2 100 K, the incommensurate-like magnetic ordering appears

to be described with a combination of the magnetic, k, and nuclear, q, propagation vectors suggesting

that the second transition takes place because longer-range magnetic correlations are established in the

alpha-like stacking sequence(s). Assuming that the same superspace group defines also the magnetic

order at T< 100 K and taking into account a Mn1-site spin-configuration similar to that of the Mn2-site,

magnetic scattering is calculated only for the k+q satellite positions. However, its relative intensity does

not match the experimentally observed one, pointing out that additional spin modulation of the existing

structure is required in order to adequately reproduce the observed magnetic NPD pattern. Rietveld re-

finements of the magnetic structure confirmed that the magnetic phase below TN2 can be described by a

proper screw component, with propagation vector k+q for both Mn1 and Mn2 sites, while refinements

assuming a spin-density wave type of structure produced worse agreement factors and unphysical mo-

ment size for the Mn1 site. The corresponding Rietveld refined 5 K NPD profile is shown in Figure 10,

with the refined magnetic parameters compiled in Table S4 27

. The associated reliability parameters are,

RFobs= 8.41% for the nuclear reflections and RFmag= 9.4% for the magnetic ones, while the RP= 16.6% is

relatively poor again due to the extreme peak broadening. The magnetic structure below TN2 is depicted

in Figure 11a-b.

To a first approximation the spin configuration is similar to the commensurate one that develops be-

low TN1, but at the “boundary” of the α- and β- like stacking sequences (Figure 5), as the ordering at the

Mn1-site (α-NaMnO2 layer stacking sequence) acts as a perturbation to the Mn2-site, the Mn-spins start

to rotate away from the commensurate structure type (Figure 9a). Within this modulated behavior, the

11

Mn3+

magnetic moment takes the lowest values within the NaMnO2 layers characteristic of the α-

polytype (likely due to their higher degree of spin-frustration), while it grows in magnitude as we move

within the β-like stacking sequences, reaching a maximum, μ 3.5(10) μB, at their mid-point (see Figure

S4) 28

. Such a non-trivial magnetic order is in line with very broad distribution of spin-lattice relaxation

times found by NMR below TN2 (cf. low-value of the stretching exponent), implying a broad distribu-

tion of local environments. This complexity might be an outcome of the system’s effort to relieve com-

peting interactions amongst neighboring spins in the β-NaMnO2 modulated nuclear structure, therefore

requiring further insights on the role of geometric frustration.

E. Parameterization of Magnetic Excitations.

Since the NPD and the NMR resolved two magnetic regimes, the magnetic fluctuations of β-NaMnO2

were studied by inelastic neutron scattering (INS). An overview of the measured INS response, well

within the magnetically ordered state (5 K), is shown in Figure 12a for experiments on the MARI spec-

trometer. A complementary insight on the low energy magnetic dynamics was offered with higher reso-

lution through the DCS spectrometer (Figure 13). At low temperatures (1.5 K and 75 K) the DCS spec-

tra show clearly the presence of a spin-gap in the excitation spectrum, with little change in the gap en-

ergy, Δ 5 meV. A pronounced change is observed at 100 K with a filling of the gap, yet with the pres-

ence of significant magnetic scattering even at T >TN1 (see Figure S5) 28

.

As the measured neutron scattering cross section is proportional to the structure factor S( , ω), for a

powder material, the measured, momentum integrated neutron intensity is proportional to the following

average at a fixed | |, )=

. Obtaining microscopic exchange interactions that form

the basis of the magnetic Hamiltonian from powder neutron data is rather difficult owing to the averag-

ing over all reciprocal space directions, | |. However, applying sum rules allows information to be ob-

tained about the interactions and correlations in a general way which is independent from the micro-

scopic Hamiltonian. We outline this method in the following.

In the absence of a full theory for the magnetic exchange interactions in β-NaMnO2, and lack of single

crystal data, we have parameterized the dispersion E( ) with a phenomenological expression which sat-

isfies the periodicity of the lattice and hence Bloch’s theorem. One possible form of the dispersion, con-

sistent with lattice periodicity can be written as a Fourier series Δ , where

is a bond vector connecting nearest-neighbor (NN) spins and Bd are coefficients in this series expan-

sion, and Δ is the magnitude of the spin-gap. Because the magnetic excitations appear relatively sharp

in energy (Figures 12, 13), we could utilize the single mode approximation (SMA) which states that the

structure factor, which is proportional to the neutron cross section, is dominated by a single resonant

mode.

The problem of deriving a parameterization of the neutron cross section, Sαα

( , ω)= S( ) δ( ω –

E( )) (delta function being numerically approximated by a Lorentzian with the energy resolution

width), is reduced to finding an expression for S( ). To do this, we apply the Hohenberg-Brinkmann

first moment sum rule, 56

which applies to the case of isotropic exchange and is closely related to the

ground state magnetic energy. Effectively the first moment sum relates S( ) to the dispersion E( ) through the following expression:

(4)

12

In view of this, the single-mode approximation and parameterization of the dispersion, E( ), allows us

to characterize which correlations are important and also determine the dimensionality of the excita-

tions. In particular, the energy gap in a powder averaged constant-Q scan is sensitive to the dimension-

ality of the interactions. This fact was previously used to show that α-NaMnO2 is dominated by one-

dimensional magnetic correlations.37

Comparison of the powder averaged spectra for β-NaMnO2 against its closely related α-NaMnO2 sys-

tem (see Figure S6) 28

points to several key differences. First, the spectral weight in α-NaMnO2 is con-

centrated at low energies near the energy gap edge, while it is much more evenly distributed in energy

in the case of the β-NaMnO2 variant. The scattering is also much more strongly peaked56

in momentum

for β-NaMnO2, which is indicative of the higher (cf. than the quasi-1D of the α-phase) dimensionality

of the associated spin correlations. In addition, considerable spectral weight is located at the top of the

excitation band and the scattering is much more well-defined in momentum than in the α-polytype.

Such qualitative observations, suggest that β-NaMnO2 may be more two-dimensional than the α-phase.

We have therefore simulated the powder averaged spectra by considering the case of the two-

dimensional spin-exchange, with dominant correlations along the bo-crystal axis. We have taken the

dispersion relation to have the following phenomenological expression:

(5)

which is consistent with the periodicity of the lattice (Pmmn symmetry) and gives a minimum at half

integer positions, relating the observed magnetic Bragg peaks. We have chosen B0= 25 meV2 to account

for the spin-gap (Δ), B1=B2= 625 meV2 and B3= 400 meV

2.

To extract an estimate for the exchange constants, we have put the inelastic magnetic response on an

absolute scale using the internal incoherent elastic line as a reference. The absolute calibration com-

bined with the first moment sum rule afforded an estimate of Jd dSS 0 . Combined with the collinear

magnetic structure, we have estimated a strong exchange along the b0, J1 = 5.0 1.0 meV and a weaker

one along a0, J3 = 1.5 1.0 meV (Figure S1) 28

.

The total integrated spectral weight (elastic and inelastic) is constrained by the zeroth

moment sum

rule which can be summarized as follows:

(6)

Integrating the INS data by using the elastic incoherent scattering of the vanadium as an internal

standard gives the inelastic contribution to the above integral being 1.8(3). Including the ordered mo-

ment in the elastic channel and noting that there are two Mn3+

ions per unit cell gives a total integral of

4.7(4) for this sum. Given the expected value for S= 2 is 12, this indicates that more than half of total

moment resides elsewhere in momentum and energy. One possibility is for a large fraction residing in

diffuse scattering, which maybe resulting in a low-energy contribution that is beyond the resolution of

the spectrometer, while it is in agreement with the broad shape of the magnetic reflections in the diffrac-

tion data and with the high density of structural defects present in the material.

F. Incommensurate Structure and Frustration.

We have seen that the magnetic long-range order of β-NaMnO2 is strongly correlated with its structur-

al complexity, which is established through the relief of frustration. Importantly, competing interactions

between spins and their complex magnetic orders are known to motivate spectacular cross-coupling ef-

fects that lead to improper ferroelectricity in frustrated magnets.57

Establishing cross-control of the

magnetic and ferroelectric polarisations challenges scientific endeavours as striking new multi-ferroic

device concepts may be realized.58

A key question then is whether the compositionally modulated nu-

13

clear structure and magnetic order in β-NaMnO2 may also stimulate competing degrees of freedom that

can become cross-correlated through the symmetries59

of the associated magnetic and nuclear order-

ings. Preliminary evidence for such a type of behavior in β-NaMnO2 was first reported by Bakaimi et

al. who demonstrated that the temperature-dependent dielectric permittivity, ε(Τ), displays two small

anomalies, near the TN1 and TN2 transitions discussed here.60

Since the explanation of possible

magnetoelectric coupling needs the understanding of the crystal and magnetic symmetries, these early

findings remained unexplored. Now that these structures are known, through the current work, it is

worth revisiting the coupling of the afore-mentioned properties.

Let us now glance through the dielectric response of β-NaMnO2 and compare it to that of α-NaMnO2.

Bearing in mind that the magnitude of the dielectric permittivity anomalies in β-NaMnO2 becomes larg-

er with the application of an intense electric field,60

here instead we utilized a progressively stronger

external magnetic field, hoping for enhanced changes in the ε(Τ). Our dielectric permittivity experi-

ments, however, identified only small anomalies in ε(Τ, H) curves that coincide with the onset of anti-

ferromagnetic orders taking place in the bulk α- (TN= 45 K) and β- (TN2= 95 K) phases. In β-NaMnO2,

no other low-temperature ε(Τ,H) signature is observed that could indicate contributions from α- and β-

type structural domains, as local probes have resolved before.41

Moreover, the magnetoelectric coupling

must be weak in both NaMnO2 materials, as very little changes are brought about despite the strength of

the externally applied magnetic field (Fig. 14). Having taken into account the symmetry-imposed con-

straints for the free-energy 61

in the α- and β- magnetic phases, it is conferred that the spatial inversion

symmetry is not violated, excluding the possibility of improper ferroelectricity in the magnetically or-

dered states (see Section S7, Supplemental Material). In this respect, it is postulated that the observed

small anomalies in the dielectric constant are likely related to the non-linear, higher order terms (e.g. bi-

quadratic term E2H

2) that are operative in chemically diverse systems, ranging from planar magnets

62,

63 and three-dimensional magnetoelectric perovskites (AMnO3, A= Y, Bi)

64, 65 to quantum para-electrics

(EuMeO3)66, 67

.

IV. SUMMARY & CONCLUSIONS.

The present work entails a thorough study of the crystallographic and dynamical properties of the β-

NaMnO2. The proposed single-phase nuclear structure model, takes advantage of the superspace for-

malism to describe the incommensurate compositional modulation (propagation vector, q= (0.077(1), 0,

0)) of the Mn and Na sites that can be depicted as an intergrowth α- and β- like oxygen coordinations.

This peculiar topology strongly influences the physical and chemical properties of the material and un-

derlines the role of the nearly degenerate in energy α and β layer stacking sequences. The remarkable

flexibility of β-NaMnO2 to adapt its lattice topology is likely at the basis of the particular high charge

capacity of the system as a Na-ion cathode material,35

but also may corroborate to the stability of the

various non-stoichiometric phases41

accessible through its electrochemical Na-intercalation/removal.68

Moreover, the magnetic structure of β-NaMnO2 was solved on the basis of time-of-flight neutron

powder diffraction data and found to be strongly mediated by the material’s inherent lattice topology.

First, below TN1 (200 K), a collinear commensurate antiferromagnetic state, involving only the β-like

stacking sequences, develops with a propagation vector k= (½ ½ ½). Then, a second magnetic transition

is observed at TN2 (95 K), marked by new satellite reflections ascribed to the interaction of k with the

compositional modulation vector q. The new magnetic ordering is due to the relief of the magnetic frus-

tration in the α-like sheets that in turn influences the ordering in the β-like stacking sequences, and in-

stigates a cooperative proper-screw magnetic state. Here, the lattice topology of the Jahn-Teller active

Mn3+

cation drives the original 3D spin correlations (T< TN1) to become 2D in character. Inelastic neu-

14

tron scattering and 23

Na NMR provide evidence that a spin-gap (Δ= 5 meV) opens in the excitation

spectra, in line with the 2D nature of the magnetic interactions at T< TN2.

Overall, structure and dynamics point that the incommensurate β-NaMnO2 structure can relay a mag-

netocapacitance effect in the low-temperature magnetic state. Such a structural complexity, inquires

whether controlled engineering of coherent defects may impart the material with novel technological

capabilities. In view of this, it is worth considering that in the compositionally modulated β-NaMnO2,

domain-wall (DW)-like phenomena 69

associated with the abundance of the α- and β- interfaces (Fig-

ures 6 and 12), rather than extended domains themselves, may be the active element in promoting some

degree of topologically correlated (related to DW), cooperative magnetic and electric dipole arrange-

ments. The way electronic structure changes at such interfacial regions could be relevant in order to

manipulate the magnetoelectric response 70

even in this class of non-perovskite compounds and war-

rants further exploration.

ACKNOWLEDGMENTS

We thank the Science and Technology Council (STFC) for the provision of neutron beam time at ISIS

Facility. Access to DCS was provided by the Center for High Resolution Neutron Scattering, a partner-

ship between the National Institute of Standards and Technology and the National Science Foundation

under Agreement No. DMR-1508249. This work was partly funded by the Carnegie Trust for the Uni-

versities of Scotland, the Royal Society, and the EPSRC. Partial funding was also secured through the

framework of the Heracleitus II project (Grant No.349309.WP1.56) co-financed by the Ministry of Ed-

ucation and Religious Affairs, Greece and the European Social Fund, European Union (Operational

Program ‘Education and Lifelong Learning’ of the National Strategic Reference Framework, NSRF,

2007-2013).

AUTHOR INFORMATION

Corresponding Author

* e-mail: lappas@iesl.forth.gr

Present Address † Ioanna Bakaimi, Department of Chemistry, University of Southampton, Southampton, SO 171 BJ,

UK.

15

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17

Figures

Figure 1. Temperature dependent (a) zero-field cooled dc magnetic susceptibility, χ(T) (right-axis), un-

der an applied field of 20 mT, and the heat capacity, C(T) (left-axis), of β-NaMnO2. The red line over

the C(T) data is the calculated phonon contribution to the specific heat, Cph(T) (see text). (b) The heat

capacity remaining after subtracting the Cph(T) contribution from the experimental C(T) depicts two

anomalies assigned as TN1 and TN2.

18

Figure 2. (a) Normalized 23Na NMR powder spectra of β-NaMnO2 revealing two different magnetic

regimes that evolve with temperature lowering. The spectra are shifted vertically for clarity. The insets

point to a specific part of the spectra, where the quadrupolar frequency is indicated by the vertical

dashed line. (b) The temperature dependence of the 23

Na NMR central line intensity multiplied by tem-

perature for β-NaMnO2. The arrows indicate the two transition temperatures TN1 and TN2.

19

Figure 3. (a) Normalized magnetization-recovery curves at a few selected temperatures. The datasets

are shifted vertically for clarity. The solid lines are fits of a stretched single-component magnetic-

relaxation model for I= 3/2 (Eq. 2; see text), while the dashed line corresponds to the fit with two such

components. Please note that significantly different stretching exponent α is found for temperatures

above TN1 and below TN2. (b) The temperature dependence of the spin-lattice relaxation rate for β-

NaMnO2. The arrows indicate the two transition temperatures. A double-component fit is needed in the

intermediate temperature regime TN2 < T < TN1. The solid lines indicate a critical type of behaviour for T

> TN2 and an activated one for T < TN2 (see text for details).

20

Figure 4. Rietveld plot at 300 K for the β-NaMnO2 structure in the Pmmn(α00)000 superspace group.

Inset: zoom of the low d-spacing region inferring that stacking faults and defects give rise to a pecu-

liarly broadened profile function. In both panels observed (black crosses), calculated (red line) and dif-

ference (blue line) pattern are shown. The tick marks indicate the calculated position of the main (black

ticks) and satellite reflections (green ticks). The asterisk marks the main reflection form the α-NaMnO2

impurity, whereas the hash-tags indicate, for example, two satellite peaks that are slightly off with re-

spect to the calculated Bragg position indicating the possibility of the other two components of the

modulation vector to be different from zero (see text for details).

21

Figure 5. Projection of the structure in the ac-plane, depicting the refined incommensurate composi-

tional modulated structure; two types of stacking changing between the NaMnO2 polymorphs are

shown. The violet atoms represent the Mn, the yellow ones the Na, and the red spheres the oxygen at-

oms. The small rectangle indicates the unit cell of the average Pmmn structure (see Figure S1).

22

Figure 6. Fourier maps of the observed structure factor (Fobs) depicting the crystallographic cation sites

(a) and oxygen positions (b). The solid colored lines represent the calculated position of the atoms

showing no positional modulation along the x4 for the Mn/Na but its presence for the oxygen sizes (vio-

let Mn, green Na, red oxygen and blue the primed oxygen position). The black continuous lines indicate

the positive density iso-surface and the dashed lines the negative iso-surface (the neutron scattering

length for the Mn atoms is negative). The iso-surface contours correspond to 2 scattering density units

(Å-2

) in all the plots.

23

Figure 7. (a) A long d-spacing section of the neutron powder diffraction patterns as a function of tem-

perature, showing the complex nature of the magnetic contribution to the pattern. Color map: the neu-

tron scattering intensity. (b) Integrated intensity versus temperature for the main magnetic reflections

with propagation vector k= (½, ½, ½), and for the satellites with propagation vector k+q, where q =

(0.077(1),0,0). The lines over the data points depict the fit to the critical region (see text).

24

Figure 8. Rietveld plot at 100 K of the β-NaMnO2 structure in Ca2'/c'(α0)00 superspace group, with

cell parameters a= 5.7108(2) Å, b= 12.6394(9) Å, c= 5.5397(4) Å, β= 120.96(7)°, and q= (0,0,

0.078(1)). Observed (black crosses), calculated (red line) and difference (blue line) patterns are re-

ported. The tick marks indicate the calculated position of the main (black ticks) and satellite reflections

(green ticks).

25

Figure 9. Sketch of the magnetic structure below 200 K, (a) along the Mn zig-zag chain typical of the

β- polymorph (ao direction) and (b) in the same projection as for Figure 6 (top panel). The black rectan-

gle depicts the unit cell of the average Pmmn structure (a0= 4.7851(2) Å, b0= 2.85699(8) Å, c0=

6.3287(4) Å), while the red rectangle indicates the unit cell of the average low temperature monoclinic

structure (am= 5.7112(2) Å, bm= 12.6388(9) Å, cm= 5.5365(4) Å, β= 120.97(7)°); please note that the

cm-axis is inclined by 60° out of the plane.

26

Figure 10. Rietveld plot at 5 K for the β-NaMnO2 structure in Ca2'/c'(α0)00 superspace group, with

cell parameters a= 5.7112(2) Å, b= 12.6388(9) Å, c= 5.5365(4) Å, β= 120.97(7)°, and q=(0, 0,

0.081(1)). Observed (black crosses), calculated (red line) and difference (blue line) patterns are shown.

The tick marks indicate the calculated position of the main (black ticks) and satellite reflections (green

ticks). The asterisk marks the main nuclear and magnetic reflections from the α-NaMnO2 impurity

phase, whereas the diamond indicates the main MnO magnetic reflection.

27

Figure 11. (a) Schematic of the β-NaMnO2 modulated magnetic structure at 5 K, projected at the same

plane as the nuclear structure shown in Figure 6 (top panel). (b) Sketch of the incommensurate part of

the magnetic structure depicting a proper screw order propagating along the (-110) direction with re-

spect to the average Pmmn unit cell. In both panels the axes directions with subscript ‘0’ indicate the

average orthorhombic Pmmn cell (black rectangle), whereas the axes with subscript ‘m’ indicate the di-

rection of the low-temperature monoclinic structure (red rectangle).

28

Figure 12. (a) The powdered averaged magnetic scattering in β-NaMnO2 and (b) the corresponding

Single Mode Approximation (SMA) heuristic model, with two-dimensional (2D) interactions. The

background subtraction method to remove phonon scattering and instrument background are described

in the text. Color map: indicates the powder average scattering intensity ) (see text for details).

29

Figure 13. The temperature dependence of the low-energy magnetic fluctuations in β-NaMnO2, meas-

ured on the high-resolution DCS spectrometer. All data has been corrected for a temperature independ-

ent background using the detailed balance relation. Color map: indicates the powder average scattering

intensity.

30

Figure 14. Temperature dependent dielectric permittivity, ε(T), as a function of the applied magnetic

field for α-NaMnO2 (a) and β-NaMnO2 (b).